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Contents
Evaluation of Algebraic Expression
Factorisation of Quadratic Polynomials
Polynomial Identities
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Polynomial
Identities
 Degree,
coefficient and constant
In the term axn,
a is called the coefficient of xn. For example, the polynomial x3 +
2x2 + 3 can
be written as 1x3 - 2x2 + 0x + 3x0.
Hence, coefficient
of x3 = 1,
coefficient of x2 = -2,
coefficient of x = 0,
coefficient of x0 = 3 (This is the constant term)
A constant
is a term with no variable beside it.
The degree of a
polynomial in x is the highest power of x. For example, the degree of x3 +
2x is 3, 2x + 8 is 1 and that of constant 5 (5x0) is 0.
Identity
Consider the
polynomial x2 -4 and (x - 2) (x + 2). They are identical, that is, they are the same thought expressed differently.
So x2 -4
= (x - 2)(x + 2) is true for all value for x. The equality is called an
identity, and may be written as:
x2 -4
(x - 2)(x + 2)
Many formulae are
actually identities. For example,
(a - b)2
a2 - 2ab + b2
Example: Given that x3 - 2x2 + 5 = ax(x - 1)2 + b(x - 1) + c for all values of x, find the value of a, b and c. Study the the formulae carefully, first, we should eliminate two of the constants. Thus by letting x = 1, we can find c. Let x = 1 1 - 2 + 5 = c c = 4 Then, we need to eliminate either a or b. By letting x = 0, we can find b. Let x = 0 5 = -b + c = -b + 4 b = -1 Let x = 2 8 - 8 + 5 = 2a + b + c 5 = 2a + (-1) + 4 a = 1 Alternatively, x3 - 2x2 + 5 = ax(x - 1)2 + b(x - 1) + c = ax3 - 2ax2 + ax + bx - b + c = ax3 - 2ax2 + (a + b)x - b + c Since LHS and RHS are identical, the coefficients of every like powers of x must be equal. x3 : a = 1 x2 : a + b = 0 b = -1 x0 : c - b = 5
c = 4
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